#  SCIENCE CHINA Information Sciences, Volume 63 , Issue 12 : 222303(2020) https://doi.org/10.1007/s11432-020-3033-3

## Statistical CSI based design for intelligent reflecting surface assisted MISO systems More info
• ReceivedJun 14, 2020
• AcceptedAug 12, 2020
• PublishedOct 30, 2020
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### Abstract ### Acknowledgment

This work was supported by National Key RD Program of China (Grant No. 2019YFB1803400), the NSFC-Zhejiang Joint Fund for the Integration of Industrialization and Informatization (Grant No. U1709219), and National Natural Science Foundation of China (Grant No. 61922071).

### Supplement

Appendix

Proof of Proposition 3.1

Applying the Jensen's inequality, we have \begin{align}{\rm E}\left\{ \log_2 \left( 1+\gamma \right) \right\} \lessapprox \log_2 \left(1+{\rm E}\{\gamma\} \right) = \log_2 \left( 1+\gamma_0 {\rm E} \left\{ \left|({\boldsymbol h}_2^{\rm T} \boldsymbol{\Phi} {\boldsymbol H}_1+\lambda{\boldsymbol g}^{\rm T}) {\boldsymbol f} \right|^2 \right\} \right). \tag{29} \end{align} The remain task is the derivation of ${\rm~E}~\{~\left|({\boldsymbol~h}_2~\boldsymbol{\Phi}~{\boldsymbol~H}_1+\lambda{\boldsymbol~g})~{\boldsymbol~f}~\right|^2~\}$. Applying the binomial expansion theorem, we have \begin{align}& \left| \left({\boldsymbol h}_2^{\rm T} \boldsymbol{\Phi} {\boldsymbol H}_1+\lambda {\boldsymbol g}^{\rm T} \right) {\boldsymbol f} \right|^2 \tag{30} \\ &= \left| \left(a_2 {\bar{\boldsymbol h}}_2^{\rm T}+ b_2 {\tilde{\boldsymbol h}}_2^{\rm T} \right) \boldsymbol{\Phi} \left(a_1 {\bar{\boldsymbol H}}_1 + b_1 {\tilde{\boldsymbol H}}_1 \right) {\boldsymbol f} + \lambda\left(a_0 {\bar{\boldsymbol g}}^{\rm T} + b_0 {\tilde{\boldsymbol g}}^{\rm T} \right) {\boldsymbol f} \right|^2 \\ &= \left| x_1 + x_2 + x_3 + x_4 + x_5 \right|^2, \end{align} where \begin{align}& x_1 = (a_2 a_1 {\bar{\boldsymbol h}}_2^{\rm T} \boldsymbol{\Phi} {\bar{\boldsymbol H}}_1+ \lambda a_0 {\bar{\boldsymbol g}}^{\rm T}) {\boldsymbol f}, \tag{31} \\ & x_2 = a_2 b_1 {\bar{\boldsymbol h}}_2^{\rm T} \boldsymbol{\Phi} {\tilde{\boldsymbol H}}_1 {\boldsymbol f}, \tag{32} \\ & x_3 = b_2 a_1 {\tilde{\boldsymbol h}}_2^{\rm T} \boldsymbol{\Phi} {\bar{\boldsymbol H}}_1 {\boldsymbol f}, \tag{33} \\ & x_4 = b_2 b_1 {\tilde{\boldsymbol h}}_2^{\rm T} \boldsymbol{\Phi} {\tilde{\boldsymbol H}}_1 {\boldsymbol f}, \tag{34} \\ & x_5 = \lambda b_0 {\tilde{\boldsymbol g}}^{\rm T} {\boldsymbol f}. \tag{35} \end{align} It is easy to observe that $x_1$ is a constant and ${\rm~E}\{x_i\}~=~0$ holds for $i~=~2,3,4,5$. Besides, since ${\tilde{\boldsymbol~h}}_2$, ${\tilde{\boldsymbol~H}}_1$ and ${\tilde{\boldsymbol~g}}$ have zero means and are independent with each other, we can derive that \begin{align}& {\rm E} \left\{ \left|({\boldsymbol h}_2^{\rm T} \boldsymbol{\Phi} {\boldsymbol H}_1+\lambda {\boldsymbol g}^{\rm T}) {\boldsymbol f} \right|^2 \right\} \tag{36} \\ & = {\rm E} \{ | x_1 + x_2 + x_3 + x_4 + x_5 |^2 \} \\ & = |x_1|^2 + {\rm E}\{|x_2|^2\} + {\rm E}\{|x_3|^2\} + {\rm E}\{|x_4|^2\} + {\rm E}\{|x_5|^2\}. \end{align} Let ${\boldsymbol~w}~\triangleq\tilde{\boldsymbol~H}_1~{\boldsymbol~f}$. ${\rm~E}\{|x_2|^2\}~\in~\mathbb{C}^{N~\times~1}$ can be expressed as \begin{align}{\rm E}\{|x_2|^2\}= a_2^2 b_1^2 {\rm E} \left\{ \mathsf{tr} \left( \bar{\boldsymbol h}_{2}^{*} \bar{\boldsymbol h}_{2}^{\rm T} {\boldsymbol w} {\boldsymbol w}^{\rm H} \right) \right\}=a_2^2 b_1^2 \mathsf{tr} \left( \bar{\boldsymbol h}_{2}^{*} \bar{\boldsymbol h}_{2}^{\rm T} {\rm E} \left\{ {\boldsymbol w} {\boldsymbol w}^{\rm H} \right\} \right). \tag{37} \end{align} Noticing that ${\rm~E} \left\{~{\boldsymbol~w}~{\boldsymbol~w}^{\rm~H}~\right\}=~\bf{I}_N$, we have \begin{align}{\rm E}\{|x_2|^2\}=a_2^2 b_1^2\mathsf{tr} \left( \bar{\boldsymbol h}_{2}^{*} \bar{\boldsymbol h}_{2}^{\rm T} \bf{I}_N \right)=a_2^2 b_1^2 N. \tag{38} \end{align} Following the similar lines, it can be derived that \begin{align}&{\rm E}\{|x_3|^2\} = b_2^2 a_1^2 \left\|{\bar{\boldsymbol H}_\text{1} {\boldsymbol f}}\right\|^2, \tag{39} \\ &{\rm E}\{|x_4|^2\} = b_2^2 b_1^2 N, \tag{40} \\ &{\rm E}\{|x_5|^2\} = \lambda^2 b_0^2. \tag{41} \end{align} Summing over all the values yields the desired result.

### References

 Wu Q, Zhang R. Towards Smart and Reconfigurable Environment: Intelligent Reflecting Surface Aided Wireless Network. IEEE Commun Mag, 2020, 58: 106-112 CrossRef Google Scholar

 Cui T J, Qi M Q, Wan X. Coding metamaterials, digital metamaterials and programmable metamaterials. Light Sci Appl, 2014, 3: e218-e218 CrossRef ADS arXiv Google Scholar

 Tao Q, Wang J W, and Zhong, C J. Performance analysis of intelligent reflecting surface aided communication systems. IEEE Communications Letters, 2020. Google Scholar

 Hu X, Zhong C, Zhu Y. Programmable Metasurface-Based Multicast Systems: Design and Analysis. IEEE J Sel Areas Commun, 2020, 38: 1763-1776 CrossRef Google Scholar

 Gao J, Zhong C, Chen X. Unsupervised Learning for Passive Beamforming. IEEE Commun Lett, 2020, 24: 1052-1056 CrossRef Google Scholar

 Wu Q, Zhang R. Intelligent Reflecting Surface Enhanced Wireless Network via Joint Active and Passive Beamforming. IEEE Trans Wireless Commun, 2019, 18: 5394-5409 CrossRef Google Scholar

 Wu Q, Zhang R. Beamforming Optimization for Wireless Network Aided by Intelligent Reflecting Surface With Discrete Phase Shifts. IEEE Trans Commun, 2020, 68: 1838-1851 CrossRef Google Scholar

 Yang Y, Zhang S, Zhang R. IRS-Enhanced OFDMA: Joint Resource Allocation and Passive Beamforming Optimization. IEEE Wireless Commun Lett, 2020, 9: 760-764 CrossRef Google Scholar

 You X, Zhang C, Tan X. AI for 5G: research directions and paradigms. Sci China Inf Sci, 2019, 62: 21301 CrossRef Google Scholar

 Ding Z, Vincent Poor H. A Simple Design of IRS-NOMA Transmission. IEEE Commun Lett, 2020, 24: 1119-1123 CrossRef Google Scholar

 Qi Q, Chen X, Zhong C. Physical layer security for massive access in cellular Internet of Things. Sci China Inf Sci, 2020, 63: 121301 CrossRef Google Scholar

 Zhang Y, Zhong C, Zhang Z. Sum Rate Optimization for Two Way Communications With Intelligent Reflecting Surface. IEEE Commun Lett, 2020, 24: 1090-1094 CrossRef Google Scholar

 Wang P L, Fang J, and Li H B. Joint beamforming for intelligent reflecting surface-assisted millimeter wave communications. 2019,. arXiv Google Scholar

 Zhang J Z, Zhang Y, Zhong C J, and Zhang Z Y. Robust design for intelligent reflecting surfaces assisted MISO systsems. IEEE Communications Letters, 2020. Google Scholar

 Taha A, Alrabeiah M, Alkhateeb A. Enabling large intelligent surfaces with compressive sensing and deep learning. 2019,. arXiv Google Scholar

 Yang Y, Zheng B, Zhang S. Intelligent Reflecting Surface Meets OFDM: Protocol Design and Rate Maximization. IEEE Trans Commun, 2020, 68: 4522-4535 CrossRef Google Scholar

 Zheng B, Zhang R. Intelligent Reflecting Surface-Enhanced OFDM: Channel Estimation and Reflection Optimization. IEEE Wireless Commun Lett, 2020, 9: 518-522 CrossRef Google Scholar

 Mishra D, Johansson H. Channel estimation and low-complexity beamforming design for passive intelligent surface assisted MISO wireless energy transfer. In: Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2019. 4659--4663. Google Scholar

 Han Y, Tang W, Jin S. Large Intelligent Surface-Assisted Wireless Communication Exploiting Statistical CSI. IEEE Trans Veh Technol, 2019, 68: 8238-8242 CrossRef Google Scholar

• Figure 1

(Color online) System model.

•

Algorithm 1 Alternating optimization algorithm

The fractional increase of the objective value is below a threshold $\varepsilon~>~0$.

$\mathbf{Output:}$ ${\boldsymbol~\phi}^{\star}~={\boldsymbol~\phi}_{i}$

and ${\boldsymbol~f}^{\star}={\boldsymbol~f}_{i}$.

$\mathbf{Initialization:}$ Given feasible initial solutions ${\boldsymbol~\phi}_{0}$, ${\boldsymbol~f}_{0}$ and the iteration index $i=0$.

repeat

For given transmit beam ${\boldsymbol~f}_{i}$, calculate the optimal phase shift beam accoring to (14), which yields ${\boldsymbol~\phi}_{i+1}$.

For given phase shift beam ${\boldsymbol~\phi}_{i+1}$, compute the optimal transmit beam according to (17), which yields ${\boldsymbol~f}_{i+1}$.

$i~\leftarrow~i+1$.

until